STOCHASTIC PROGRAMMING SOLUTIONS TO SUPPLY CHAIN MANAGEMENT

Transcription

1 STOCHASTIC PROGRAMMING SOLUTIONS TO SUPPLY CHAIN MANAGEMENT a dissertation submitted to the department of management science and engineering and the committee on graduate studies of stanford university in partial fulfillment of the requirements for the degree of doctor of philosophy in operations research Rene C. Schaub March 2009

2 Copyright c 2009 by Rene C. Schaub All Rights Reserved ii

3 I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Gerd Infanger (Principal Advisor) I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael A. Saunders I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Yinyu Ye Approved for the University Committee on Graduate Studies: iii

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5 Abstract We formulate a finite-horizon multistage stochastic linear program model for tactical supply chain planning problems and study the effect of random demands on expected profit, through appropriate material flow and inventory allocation. The associated deterministic model is a capacitated Leontief substitution system, which itself is an instance of a capacitated Hypergraph. We find a fast algorithm to compute approximate solutions, based on Positive Linear Programming. We explore different decomposition approaches and suggest enhancements to the Benders decomposition method. Supply chain test cases from the literature are used to generate models with large numbers of scenarios and stages, for which we compute near-optimal solutions. v

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7 Acknowledgments I would like to thank my advisor Prof. Gerd Infanger for his guidance and advice that always turned out to be to the point. He showed much patience for my deliberate branching into areas that I thought merited investigation. I would like to thank Profs. Michael Saunders and Yinyu Ye for their numerous advice and contageous optimism. Profs. Serge Plotkin and Warren Hausman agreed to serve on my defense committee. I could not have hoped for a better matched group of experts to present my findings. My girlfriend of many years Galicia Vaca could not believe that I finally finished this project of mine, and I am grateful for her support through all this time and hard work. Sometimes I think I may not have been able to find the endurance to finish my dissertation without her. I would also like to thank my parents for always offering support to all my endeavours, and the companies that provided me with consulting and work opportunities, providing financial independence and the ability to go where I wanted to be. vii

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9 Contents List of Tables, Figures, and Algorithms xi 1 Introduction Finite Horizon Aggregation Continuous Variables Stochastic Programming for Supply Chain Management Productions as a Directed Hypergraph Model Expressiveness of the Production Model Demand Distributions Unknown Trend Models Forecast Update Models Leontief Systems and Extensions for Capacities and Profits in the Deterministic Case Solving the Capacitated Acyclic Leontief Substitution System Minimum Cost Trees Running Times Stochastic Supply Chains using Positive Programming and Scenario Aggregation Decomposition Non-Smooth Optimization Using a Collection of Subgradients Cutting Plane Method Bundle Methods Some Stochastic Decomposition with Subgradient Optimization Algorithms Degeneracy and the Cutting Plane LP Scenario Aggregation with Bundle Method Stochastic Supply Chains Using Multistage Benders Decomposition Benders Decomposition ix

10 5.2 Multistage Benders Decomposition Multiperiod Cuts Shared Cuts Serial Correlation of Demand Capacities Vector Autoregression and Multiplicative Dependencies Positive Demand Correlated Demand Computational Results and Comparison Benders Implementation Test Supply Chains Demand Distributions Number of Stages Lower Bound Accuracy Redundant Solution Space Conclusions Bibliography 52 x

11 Tables, Figures, and Algorithms Tables 6.1 Origins of the selected supply chains Test case parameters and numerical results Descriptions of columns of Table Figures 2.1 A simple bill of materials Productions over time Cutting plane iteration Bundle method Bundle ɛ-descent method Bundle method with level sets iteration Test Case Test Case Test Case Test Case Test Case xi

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13 Chapter 1 Introduction Operational or tactical supply chain planning problems (TSCP) have a horizon of less than one year, and have long been the domain of Material Resource Planning (MRP), see [33] and [52] for historical surveys. The decisions to be made concern material flow, stocking, sourcing and timing [17, 39, 58]. Because of the short horizon it is assumed the supply chain network itself has already been designed and built; that is, manufacturing plants, distribution centers, and potential transportation routes already exist. MRP uses rules to make decisions. When there are many suppliers, a sourcing rule may allocate each one a fixed percentage of business. When an order is confirmed and material released, the planning process takes one step at a time in looking up which materials go into the current assembly, and allocates them at the earliest availability, or creates new requisitions, which in turn are followed upstream in the supply chain until all components have been accounted for and resources allocated. Kanban is a very simple, just-in-time (JIT) manufacturing planning system popularized by Toyota. Contrary to the above push -based system meaning orders can result in coordinated material decisions affecting many stages of the chain all at once Kanban is a pull -based system. Material orders are only released when a certain number of units have been used up (Kanban cards), and only for the immediate upstream stage. That way, material is pulled down the chain. The advantages are that very little coordination is required, implementation is easy, and decisions can be made very quickly. The process of setting the right number of Kanban cards is made in advance and is itself an independent planning process. What has crystallized in recent years in the commercial space is a focus on more and more integrated multi-echelon planning (enterprise or inter-enterprise wide) as opposed to the traditional decentralized local approaches as it has become clear that there are additional cost savings to centralized inventory planning once uncertainties are made part of the model [55, 36, 19]. This requires that all of the enterprise data be accessible from a central location. The enabler of such integrated data storage in practice are so-called Enterprise Resource Planning (ERP) systems [34]. During the last decade, operations research optimization techniques have been ap- 1

14 2 Chapter 1 Introduction plied to commercial MRP software, typically going under the name of Advanced Supply Chain Planning or MRP II. For early approaches see [4]. One typically sees a combination of linear programming and constraint programming. More recently, some approaches have considered uncertainties in the supply chain. Here we are mainly concerned with demand uncertainties. We propose a model that optimizes over multiple items and multiple stages ( echelons ), globally over a certain number of time units. While it may seem that a multi-echelon view spanning production and distribution would make the model too unwieldy, we build on simple constructs that allow us to take advantage of structural properties. Furthermore we restrict ourselves to continuous variables to make the model computationally tractable. For an overview of modeling approaches and issues, see [16]. Related issues that we do not address here are integral constraints such as used in the lot-sizing problem, or designing the supply chain. The latter can be solved separately and in advance of the TSCP, while the former can be treated as a post-processing step. 1.1 Finite Horizon We use a finite-horizon formulation. It can be applied for seasonal demand that vanishes after the last stage, or over the life cycle of a family of products. An infinite horizon or static model can be approximated by making the horizon sufficiently long and rolling over the time window after some number of periods. 1.2 Aggregation The length of a period is typically chosen to be between a day and a month. This implies aggregation of all time-related entities of the supply chain, such as demand amounts, resource capacities, and rounding of production lags to time periods. Clearly, accuracy is lost, and optimization of such a model may require post-processing of the solution to enforce constraints locally at a finer level. In the extreme case, aggregation results in all productions finishing in the same time bucket. The only productions with non-zero lag are the inventory productions, carrying over stock to the next period. Demand is satisfied instantaneously. The only remaining issue is material stocking levels if production cannot meet demand spikes. On the other end, we have a continuous time spectrum with individual orders and exact production lags. A compromise is struck to balance computational and other effort with accuracy.

15 1.3 Continuous Variables Continuous Variables Our model is a continuous one, with decision variables taking values on the real line. We don t include binary or integer variable constraints, which result in problems that in general cannot be solved to optimality (NP hard). The main victims are setup costs and lot sizing, which, however, become progressively less important at higher levels of aggregation. One possibility for approximately modeling setup costs is to introduce multiple bills of materials with different lag times and capacities, corresponding to the length of production runs. Each bill will have the setup cost added as part of the total cost. The item output of the BOM can also be spread out evenly. The longer the lag, the lower the per-unit setup cost. It is not an entirely accurate way to measure setup costs, because the total amount being built is a continuous variable and may be less than capacity. The idea is that committing for a long run has a much higher risk of producing too much or too little, if the long lag is used to reduce setup costs only. As our model is capable of optimizing such profit risks, it will select shorter runs with higher setup cost if responsiveness to changes in demand is important.

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17 Chapter 2 Stochastic Programming for Supply Chain Management 2.1 Productions as a Directed Hypergraph Model Let us start by looking at a simple bill of materials (BOM), which describes the composition of an item made from raw materials or intermediate items. c a b Figure 2.1 A simple bill of materials We assume that Figure 2.1 means that to make one unit of item c, we need one unit of item b and one unit of item c. We can express this relationship as a hyperedge e E in a hypergraph (E, V ), with the items as nodes V. We adopt here the notation in [8]. A hyperedge e differs from an edge in a regular graph in that it can connect more than two nodes. e is a set {v 1,..., v k } V of the nodes. We are interested in directed Hypergraphs that have at most one destination node that is, one produced item denoted by h e (head of edge). The set of source nodes is denoted by T e (tail of edge). Then the hyperedge e is the pair (T e, h e ). In our example, the bill of materials is hyperedge e, the produced item is h e = {c}, and the components are T e = {a, b}. As the components used may not be exactly one unit each, we have the usage p(v, e) indicating the amount. The number of time units to finish production is indicated by l e. Costs associated with unit production of h e are denoted by c(e). Given these parameters, we now introduce a decision variable x e for each production in a given time period to indicate how many units are being produced. To summarize, 5

18 6 Chapter 2 Stochastic Programming for Supply Chain Management (E, V ) Productions e E are hyper-arcs between items v V T e h e p(v, e) l e Set of component items consumed by production e Item being produced by e Amount of item v T e h e required ( 0) or produced ( 0) in production e Time units to complete production e (lag) c(e) Unit production cost ( 0) or revenue ( 0) x e Decision variable of how many units to produce What is interesting about our production construction is that while mainly intended to model BOMs, it actually suffices to model the whole supply chain. This implies that a solution is simply a flow on a directed hypergraph. When written as linear program equations, the corresponding production matrix is a Leontief substitution system, which has at most one positive entry in each column. We now show what we mean by the statement that we only use productions. Let us start with an inventory type equation, which keeps track of the net inventory position of item v at a given time, which we shall call I v : I v = v h e x e v T e p(v, e)x e. Note that if we allow p(v, e) to be negative or positive according to v corresponding to a component item or the item produced, we only need one summation term with v h e T e : I v = p(v, e)x e, e E(v) where E(v) E is the set of hyperarcs that contain v. We further assume that we have a distinct item v for each time period 1,..., T ; that is, v only exists for that time period. In other words, V = V 0 {1,..., T } for some initial timeless item set V 0, and similarly for E. We don t signal the time period with an index on v, rather we use t(v) and t(e) to retrieve the item s location in time, and the production s starting period, respectively. What is needed now clearly is linking the inventories I v for consecutive time periods, as otherwise any positive inventory I v simply expires. Thus we have I v = p(v, e)x e + I pr(v), e E(v) where pr(v) is the item instantiation during the previous time period, so that t(v) = t(pr(v)) + 1. Similarly, let nx(v) be the item in the next period. Now it is obvious

19 2.1 Productions as a Directed Hypergraph Model 7 that I v is simply another production e a, with T ea = {v}, h ea = {nx(v)}, p(v, e a ) = 1, p(nx(v), e a ) = 1, t(e a ) = t(v). x ea then indicates the inventory at time t(v). t = 4 D P PD t = 3 D B C PD P PB P t = 2 B C A PB P t = 1 A Figure 2.2 Productions over time. The angled icons represent unit lag hyperedge productions, linking components and output items over time periods Our inventory (or flow conservation) equation now takes the simple final form p(v, e)x e 0 (2.1) e E(v) using only productions. See Fig. 2.2 for an example bill of materials tracing each item for two time periods. For several reasons, we use greater than instead of equality in (2.1). This says that we can drop any inventory at any given time. While this would be very unlikely to occur in any optimal feasible solution, it is very useful for feasible, non-optimal solutions encountered during problem decomposition. It prevents subproblems from ever becoming infeasible, thus saving us from introducing relief inventory

20 8 Chapter 2 Stochastic Programming for Supply Chain Management productions, or dealing with feasibility cuts. It is also natural in the case of resource productions, as a time resource not used is a missed opportunity, but there is no point in forcing its usage, and it cannot be recovered by carrying it over to the next period. Each production variable can have an upper bound ū(e), and is always non-negative. In general, this corresponds to capacity constraints on processes modeled by productions. In particular though, it also allows us to model demand, for which we appropriately introduce demand productions e d, with h ed =, T ed = {v}, p(v, e d ) = 1, and ū(e d ) = d, where the capacity bound d indicates how much demand exists during the time period. Of course the incentive to satisfy demand, that is invoke the demand production decision variable x ed, is missing. We address this by adding the profit objective, c(e)x e, (2.2) e E where c(e) is the positive cost or negative revenue, so that (2.2) is to be minimized. For demand productions, c(e d ) is always negative to indicate revenue opportunity. 2.2 Expressiveness of the Production Model Now we are ready to show that the remaining essential constructs of a supply chain can also be modeled by productions. Standardized definitions can be found in the APICS dictionary [10]. Safety Stock Safety stock is just the inventory production e a. The production amount indicates how much surplus was carried over to the next period. (Safety stock trades off lead time. If we always keep enough safety stock at a location to satisfy actual local demand, the effective lead time is zero). External Supply Raw material or starting components have to be sourced from somewhere. These productions e s don t use any other items, that is, T es =. Resource In addition to using common items, productions can use common resources such as work hours available, or a shared machine. Like external supply, the resource production has T er =. In addition, there is no inventory: we drop the carry over productions e a corresponding to the resource item. Either the hours of work available are used, or they expire. Of course every resource has a limiting upper bound capacity ū(e r ). To indicate use of a common resource e r, any production e can add the resource item h er to its components T e with appropriate usage p(h er, e,).

21 2.2 Expressiveness of the Production Model 9 Warehouse Capacity and Cost If shared, limited storage availability can be modeled as a resource used by the inventory productions e a. If limited space is allocated for an item individually, we can simply set the capacity of the corresponding inventory production. The cost is charged to the respective resource or inventory production. Warehousing costs are also incurred for items in the production pipeline, proportional to the amount being produced, in which case we add the cost to the production itself. Optionally, some other concepts can be implemented using productions. Some only become meaningful in the stochastic setting, in particular when timing of decisions is concerned. The cost typically is an increase of part of the problem size by a constant multiple. Shelf Life Some types of item, for example produce or consumer electronics, expire after a while and become more or less worthless. We assume that after a certain time period since production, the item has no more use and its inventory is scrapped. We model this by introducing a version of the item indicating its current age in 1,..., exp; that is, we go from v to {v 1,..., v exp }. Instead of the usual inventory productions e av having the same item in the next period as their output, they point to the adjacent item at age +1. v exp does not have any inventory carried over, and its stock will have to be used up or expire. In addition, we have to replicate all productions feeding off v in a similar fashion so that they can use any aged version of the item. Lag It is possible to model lags > 1 using only unit lags. While it may not be clear why we would want to do that right now, we shall indicate the procedure here. Assume the production e takes l e periods to complete. Then we proceed similarly to the case for shelf life, by introducing aged versions of the output item h e. Only the last version v exp with exp = l e is used by any productions feeding off v. Backorder We have so far assumed that demand has to be satisfied during the time period in which it materializes. This corresponds to a service time of zero, which we would see for example at a consumer outlet, where the customer expects the item to be stocked and available. Similarly, it could model a situation where an order is taken and expected to be shipped during the next business day. If however we allow for a service time to fulfill the order (not counting shipping), we can model that as a backorder. That is, we allow demand to live on for a certain number of periods, until it expires if not satisfied. To do this, we introduce an

22 10 Chapter 2 Stochastic Programming for Supply Chain Management external supply that injects a shelf life token item, up to the amount of the demand during that period. This item is then passed on as far as it hasn t been used by appropriate demand productions up to the expiration period. Alternatively, we can approximate a positive service time slack s modeled as early demand, see below, without tokens. We use a stage-dependent additive distribution where the demand has a regressive lag equal to the demand slack, and demand times are moved out by s periods. Early Demand In some situations, precise demand may be assumed known in advance of the actual time of commitment to fulfill the order. This is only a slight variation of the backorder construction. The only difference is that demands may feed off the last token item only, instead of all of them. But we don t need to implement early demand that way, as it is more natural to represent it directly in a VAR demand model, see section 5.5. (Conversely, any positive VAR distribution can be implemented using token flows to carry random outcomes to future stages.) On the other hand, if early demand means early commitment, then the proper thing to do is to allocate all materials needed at the time of the order. This can be achieved as follows. Let L be the time allowed to fulfill the order. From the demand item as root, we collect all upstream production trees that have a max total lag of no more than L. Then we roll up the bills of materials so that for each tree we end with a cumulative bill of materials that has no output and as inputs all the leaf components in the tree. Depending on the supply chain and L, there may be too many such trees, in which case we could instead use the uncommitted early demand approach as an approximation. Another useful case for early demand is when a manufacturer or distributor shares demand data with an upstream supplier. The latter will know in advance what amounts are needed. Conversely, if that information is not shared, the supplier will have to create a demand forecast based on past history and in general carry greater safety stocks, or quote a longer lead time. Withholding demand information from upstream suppliers can result in greater variation of demand, the so-called Bullwhip effect [30]. Diminishing Marginal Revenue Diminishing revenue can be modeled by linearizing a concave revenue curve and splitting it into linear parts, each of which becomes a different demand production for the same item, each subsequent one with lower revenue. The same can be used for convex costs. Supplier Failure and Random Yield AMR Research has reported that businesses indi-

23 2.2 Expressiveness of the Production Model 11 cate supplier failure as their number one risk factor [23]. Supplier failure can be modeled as random capacities for supply productions. While the risk of an individual supplier failing may be small, the overall probability of at least one supplier failing will be much larger in a supply chain with many external components. Some of the risk may lie with supplier stockouts, but more dangerous are orders with long lag time that fail to arrive. To model disruption, we introduce a dummy item for each period until delivery, and put zero capacity as a possible outcome on some of the carry productions. To model recovery of supplier payments, we add an artificial, parallel demand production at cost, whose capacity is different from zero only if disruption was the outcome. The cost of this manoeuver is that outcomes with large jumps (such as zero vs non-zero capacities) generally are so different that Benders cuts (as described in chapter 5) derived on one path may not provide much support for other paths with different outcomes, so that more cuts are needed for convergence. Random yield can be modeled the same way. Alternatively, it can be implemented directly by making the output coefficient of the production matrix random. If serial random correlation is required (we only support random rhs for serial correlation), we can model it indirectly through capacities, as follows. For each outcome, we replicate the production, and set the yield according to the outcome. Randomness is then moved to the production capacities, with only one production having non-zero capacity at each outcome. Random Variable Lead Time Globalized procurement is one source for greater variability of lead times. The latter can be modeled by sending the supply into a dummy holding item each period, with connecting carry production, up to a maximum lag period. For each holding item, there is a delivery production that moves the dummy to the real item. Its capacity is determined according to the lead time distribution (in parallel, the dummy carry production may be penalized by at least the carry cost so that the supply is pushed into the real item). We can model any lag distribution by independent binary random capacities on each dummy delivery production. Cost Risk Cost increases cut into profitability. We model the risk within our existing framework by using several supply productions with different costs. A high cost scenario is characterized by limited availability of the lower cost productions. Sequential dependency of the capacity distributions makes sense here (section 5.5),

24 12 Chapter 2 Stochastic Programming for Supply Chain Management otherwise a high cost will most likely revert back by the independence of the stage random variables. Tax uncertainty from country to country in a global network also translates into cost risk and can be modeled as such. Overtime We copy the production or resource that is going into overtime and add overtime costs to its objective. Then the overtime production will be selected only when the regular production has been exhausted. Product Substitution Clearly our model allows for substituting the same item from different sources or productions. Similarly, different items can be modeled this way if they effectively can satisfy the same demand. When items are a little bit more differentiated, we would like to have separate demands, but correlated in some way. For example, an item may be a substitute for another, if the price is brought down sufficiently to make the consumer consider the lower priced item. Similarly, a customer may choose to upgrade to a more expensive item if the price is brought down sufficiently. For effect of pricing on substitutability, see for example [32]. In either case, we achieve demand pooling, thus preventing stockouts in one item to result in a lost sale. This can be easily implemented as follows: in case of the expensive item, let it flow to its original price point demand, as well as to a separate demand where it is the substitution item. The latter will have lower revenue, so the higher cost item will only satisfy that demand after there is no more of its own proper demand, and after the low cost item supply has been exhausted. Similarly for substituting with a lesser item. The opposite case is when we have items that are not substitutes, but demand is inversely correlated, in that one of them will be much more popular than the other, no matter what pricing points are chosen. Instead of risk pooling, we increase risk by differentiation. Such a case can be modeled with an appropriate, autocorrelated demand distribution. We would expect that as a profit maximizing measure, the model would decide to build sufficient inventory at intermediate assemblies before the items are differentiated. Expediting To satisfy demand expediently, sometimes it pays to expedite movement of material and products. This can be modeled as alternate transportation routes with smaller lead times and higher cost. Lastly we describe some concepts that cannot be modeled effectively in the production model and haven t already been mentioned.

25 2.3 Demand Distributions 13 Satisfaction Level One customer satisfaction or service metric is to guarantee a minimum percentage of orders filled. While this can be as simple as an upper bound on satisfying demand in a period, even in this case we are changing the model in that subproblems in Benders decomposition (chapter 5) can now be infeasible, requiring infeasibility cuts and introducing other difficulties. When the percentage is to be over multiple time units, more complications arise, which in any case invalidate our simple productions model. The good news is that satisfaction levels have been mainly in use to drive stocking levels of a single-echelon system, which makes sense in such a simple model. But in a multi-echelon, multi-item model such as ours, where we can finally begin to talk about expected profit, maximizing the latter is a lot more useful. If we are concerned about customer goodwill, we can simply assign a cost of unserved demand to each order, and add it to revenue. Then the expected lost goodwill can be broken out from the solution and reported. Desired higher order fill rates for distinct item categories (A,B,C [10]) can also be reflected in higher lost order values for the higher priority items. This seems to be a better way to account for customer satisfaction than strict percentages, which may or may not make economical sense (and certainly are not profit driven). The other case where satisfaction levels are useful is when cost data is very inaccurate. This is quite a common occurrence in ERP databases, and the culprit usually is that the cost data has not been used in any critical function, e.g. to drive profit. Of course the remedy is to improve accuracy of the costing data. Rolling Inventory A company may decide to store products in truck trailers at the back of the warehouse instead of at the warehouse itself [47]. As this is trying to maximize individual trailer usage, it is better to model this as a MIP and thus falls outside the scope of our model. 2.3 Demand Distributions The question of what kind of demand distribution (expressed as the values ū(e d )) we should have needs to be addressed. Demand should be non-negative. Negative demand can be interpreted as customer returns, although realistically returns are a small fraction of demand and would be very unlikely to outnumber total sales during a given period. Models composed of normal random variables can always be negative, albeit

26 14 Chapter 2 Stochastic Programming for Supply Chain Management the probability can be kept arbitrarily small when the mean is sufficiently large and the variance small. When errors are not normal, more safety stock is usually needed [54]. A variety of standard non-negative distributions lends itself as building blocks to time series construction. We shall not elaborate on conditions for stationarity, which is extensively treated in the literature. A stationary model is easier to estimate. However, even a trend or seasonal adjustment does not hide the fact that variance remains constant over time. We believe it is more natural for the demand variance to grow with time, rather than to assume a fixed underlying trend. The toolbox of standard continuous non-negative distributions includes the exponential, truncated normal, conditional normal [54], lognormal, Weibull, and Gamma distribution. For the positive integers, there are the Poisson and binomial distributions. Simple autoregressive models with non-normal innovation have been analyzed in [20]. Correlation between time series can be captured in a (lagged) covariance matrix. For a vector normal distribution, this completely determines all parameters. By conditioning on positive vector outcomes (i.e. discarding sample vectors with negative coefficients), we obtain a model that may be easy to fit and allows for negative autocorrelation. The probability of a negative outcome should be small however, otherwise conditioning may be hard to achieve. Negative correlation can also be easily realized without allowing the vector to go negative by correlating the positive vector of innovation in a multiplicative or additive vector autoregressive model (fitting the model may have its own challenges, but we ignore that problem here). However we do not see the need to model negative correlation: the main case that comes to mind is cannibalization, which seems to be avoidable when the supply chain in question is for products of the same company. Instead, we would expect the products to be substitutable, in which case a common demand distribution can be found for the product family of substitutes. Similarly, introducing a new product replacement does not need to be left to random chance, a strategy might be to force production capacities for the replaced item to go to zero quickly. We do not address the question of how to estimate or select among the various models given historical samples and other information Unknown Trend Models For illustrative purposes, we chose a model that exhibits positive autocorrelation but no correlation between demand time series, and has growing variance. (The model can be easily extended to dependencies between time series, for example by sharing some of the underlying variables or series.) one instance of our model exhibits a linearly

27 2.3 Demand Distributions 15 growing standard deviation using a triangle distribution construction (with some noise thrown in if desired to add another source of variability). We believe this to be a very good approximation of actual uncertainty, as when demand peaks is the most critical unknown given a certain growth trend. We acknowledge this very important source of uncertainty by using a distribution that actually reflects it. To hedge, we have to know what we are hedging against. In practice the peak will announce itself by flatlining growth close to it, but triangle trends with common growth rates are a useful approximation. Note. If each trend outcome had a distinct incline rate, then the full triangle could be identified as soon as the gradient became visible during the first few periods. Hence we assume here that all possible trend outcomes have the same incline, so that the curve is identified only after its peak. This would not be very useful for traditional forecasting applications, as we cannot make any reliable forecasts until the peak occurs, but it is very useful indeed to let the stochastic program hedge against all such trend outcomes. Demand for a given item at stage t is defined to be d(t) = t j=1 ε jt t µ k, (2.3) k=1 where ε jt and µ k are independent non-negative random variables, for example truncated normals, and t = 1,..., T. If we let µ t be binomial random variables, and let ε jt be identical constants, we obtain the triangle distribution. It embodies the unknown peak property, as the first failure outcome µ t = 0 signals that the peak occurred at t 1. Of course ε jt can be any function of j and t, so that the individual peak demand curves can take any shape. As a second more practical instance, let µ t be some random variable with E(µ t ) > 1, Var(µ t ) < 1. (2.4) We can interpret ε jt, t = j,..., T as early demand information that becomes available in period j. Future demand at t is positively correlated with early demand information via µ k, k = 1,..., t. This implements the notion that the amount of early demand is a good predictor of future demand Forecast Update Models Stefanescu et al. [53] look at the problem of estimating a correlated demand model as data becomes available. They assume that a common initial shock vector determines

28 16 Chapter 2 Stochastic Programming for Supply Chain Management all the individual demand trends. As more data becomes available, the parameter estimation is refined. We shall now explain why this and similar models are not going to be very useful here, even when they are a good fit. The reason is that forecasts are updated as more information becomes available, and production adjusted accordingly, instead of hedging against possible future outcomes. A Bayesian approach seems conceivable, where the conditional distribution model is represented by the posterior. However this is circular, as we would have to simulate a partial outcome from the true distribution to obtain the posterior. Instead of updating a forecast or parameters for a fixed-trend model, the partial demand outcomes should be used only to condition the distribution. If we knew the exact parameters, then the stochastic program would know the trend functions and be able to optimize for the remaining random noise. Since we don t know the exact trends in advance, the trend curves can take various forms that become clear only later on, so it is a mistake to think that the stationary random noise remaining will somehow obscure the trends and lead the stochastic program to hedge against different trend curves: the conditional future distribution is fully known to the stochastic program for any simulated outcome. Any trend that becomes deterministic after some simulated point in time is fully visible for all future periods, and no hedging occurs, except for the remaining yet to be realized noise. The appropriate analogy for our model of the approach used in [53] is 1. Solve the stochastic program. 2. Implement the first stage decisions. 3. Observe the next stage demand outcome and any other visible variables. 4. Update the demand distribution given the new information. 5. Roll the time horizon window forward one period.

29 Chapter 3 Leontief Systems and Extensions for Capacities and Profits in the Deterministic Case Looking at the linear program made up of the minimizing objective term (2.2) and the item constraints (2.1), we see that the matrix P := ( p (v, e) ) is pre-leontief [56]: V E any production e is represented by a column in P that has at most one positive entry for the output item. All other items v in e are consumed and have negative p(v, e) coefficients. As we allow more than one production to output the same item (that is, we allow multiple sources), the corresponding linear program is a pre-leontief substitution system (LSS). Leontief systems, not surprisingly, have their origins in production planning and economics. For some models that are very similar to ours, see for example [11, 57]. Koehler et al. suggest solving a general LSS with matrix iterative techniques in [29]. Dantzig [11] shows how to solve efficiently the special block triangular structure resulting from a time-phased LSS with the simplex algorithm. When constructing (V, E) from time-phasing items and productions using lag l e, the matrix P takes the form A B 1 A.. B.. 1 B L., (3.1) B L B L... B 1 A where L = max e l e, P i. denotes items at time i, and P.j productions starting at time 17

30 18 Chapter 3 Leontief Systems and Extensions for Capacities and Profits in the Deterministic Case j. The band block structure is due to ordering the item rows by time t(v), and the production columns by the production starting period t(e). Alternatively we could have ordered the productions by finishing period, that is t(e) + l e, but we need to group variables by time of decision (starting period) for the stochastic version later. We don t allow cycles in the supply chain. An item may not be fed to one of its component productions. Such occurrences would mainly be seen when reverse supply chains for returns and disassembly are integrated into the main supply chain. While reverse supply chains are certainly important, we don t see a need to integrate them into the global supply chain, as the reverse flows are only a small fraction of the total. It seems sufficient to treat the reverse supply chain independently as a separate problem. We thus consider only acyclic LSS (ALSS). Based on [11], Veinott shows in [57] how ALSS can be solved in a single pass through the (ordered) data. None of these approaches applies when there are upper bound capacity constraints on the productions (as the problem is not a LSS anymore, except in special cases, see [57]). Cambini et al. [8] propose a network simplex-inspired method for solving flows on capacitated directed hypergraphs, or equivalently, capacitated Leontief substitution systems (CLSS). There is a correspondence between spanning hypertrees and bases, but unlike the network simplex method, updating the basis tree potentially involves a quadratic effort. Note that any linear program with upper bounds on all its variables can be translated into a CLSS [8]. The problem that we are interested in solving is a capacitated, acyclic Leontief substitution system (CALSS). This is the linear program min cx P x 0 (3.2) x ū, where ū and c are the production capacity and cost vector, and P is as in (3.1). 3.1 Solving the Capacitated Acyclic Leontief Substitution System Since the problem matrix P grows linearly with the number of periods, it can become quite large. As deterministic subproblems can be used to solve a stochastic version of (3.2), it is important that the many inner iterations involving (3.2) be solved quickly. In the following we take advantage of the structure of CALSS to develop an approximation

31 3.1 Solving the Capacitated Acyclic Leontief Substitution System 19 algorithm. A hypertree is a subgraph of (V, E) without undirected cycles. Theorem 3.1. Every optimal solution to CALSS (3.2) is a sum of at most E flows on hypertrees. Proof. A feasible solution can have surplus at any given item and time. But since the only productions with negative cost are the terminal demand productions, any feasible solution with discarded surplus can be transformed into a solution without surplus and equal or lower total cost. Assume optimal solutions x don t have surplus. Then P x = ū, which acts as flow conservation constraints, so that x is a flow on the hypergraph (V, E). All flow terminates at the demands. Since the hypergraph is acyclic, we can start at any demand production variable e d and trace the flow upstream. At each upstream production, we determine how much has to flow on it to satisfy the downstream flow value. If there are multiple upstream productions providing flow for a component item, then we arbitrarily pick one source e. We potentially reduce the flow if x e is not large enough. In the end, we have traced out a hypertree with the demand production e d as root, and a corresponding flow that feeds a portion or all of the demand production needs. Since the flow conservation constraints have to hold, we can subtract the tree flow from x and still have a feasible flow left. Each time we reduce at least one edge of x to zero. As there are only a finite number of productions, eventually we will have partitioned x into flows on hypertrees with demand productions as roots. We now reformulate (3.2) in terms of hypertrees. Assign each possible hypertree δ with some demand root a flow variable f(δ). The size of f indicates how many units of demand are being satisfied by the tree. The tree is represented by the vector p(., δ), indicating how many units of capacity the tree needs for each of its productions, to satisfy one unit of demand. For example, if a unit of the demand item can be made by a production using 2 units of capacity, and that production in turn has a component that can be made by another production e with 3 units of capacity, then the usage p(e, δ) of that production is 6. d δ E is the tree s demand production, c(d δ ) is the revenue of the tree s demand production, c(δ) denotes the cumulative production cost on the tree to satisfy one unit of demand. The resulting problem is max f (c(d δ ) c(δ)) f(δ) δ s.t. p(e, δ)f(δ) ū(e), e E, e δ (3.3) f 0. There can be an exponential number of possible trees, and thus solving (3.3) directly may be intractable. But there are only linearly many trees in the solution. A column

32 20 Chapter 3 Leontief Systems and Extensions for Capacities and Profits in the Deterministic Case generation type approach is needed. We need to be able to find a tree quickly given some reduced-cost type criterion. What is interesting about (3.3) is that p(e, δ) and ū(e) are non-negative, and f 0. We are thus dealing with all positive constraints, that is with a positive linear program (PLP), with only fractional packing inequalities ( ). Fast approximation algorithms exist for this problem, for example [61]. A variety of special cases are treated in [18, 40]. PLP also goes by the name of (mixed) fractional packing and covering in the Computer Science literature. Similar algorithmic approaches can also be applied when the positive constraints are nonlinear convex and concave functions [27]. What both algorithms by Young [61] and Koenemann [18] have in common is they assign dual weights to the rows and require finding the minimum weight [18] or negative weight [61] rows in an iteration, and maintain weights to remain exponential in the constraint violations. Koenemann considers a PLP of the form max cx subject to Ax b, x 0, (3.4) x and defines column j s cost as c(j) := π A.j /c j (3.5) for some π. We cannot apply (3.3) directly here, as c(j) depends on which tree has been selected for a given demand, and we don t know how to find efficiently a smallest cost tree when the cost involves a divisor that is a function of the tree. Instead, we use a budget constraint, as in [40]. The budget constraint c(δ)f(δ) β (3.6) δ separates the cost from the revenue in the profit objective of (3.3), resulting in problem g(β) := max f c(d δ )f(δ) δ subject to p(e, δ)f(δ) ū(e), e E, e δ c(δ)f(δ) β, δ f 0. (3.7) The revenue of a tree for a given demand does not depend on the selected tree, and as shown below, we can efficiently find the smallest cost tree. The budget constraint (3.6) sums up the total production cost of all trees, and requires it to be less than

33 3.1 Solving the Capacitated Acyclic Leontief Substitution System 21 some constant β. If β is too large, it means we are ignoring production costs and only maximize total revenue. If β is set too small, potential revenue is held back. For a given β, total profit is at most total revenue minus β (If the budget is not binding at an optimal solution, it implies that the trees maximizing total revenue are all profitable. Furthermore, the trees maximizing flow also use the cheaper production sources. While this is highly unlikely by chance, it certainly holds when the supply chain problem is posed pre-sourced, that is each item is only made by one production.) 1 A bisection line search can be applied to maximize g(β) β, (3.8) which is a concave function of β Minimum Cost Trees For simplicity assume that p(h e, e) = 1 for all productions. minimum cost of any tree producing item v: Designate by w(v) the w(v 0 ) = min h e=v 0 π e + c(e)π b + v T e p(v, e)w(v), (3.9) where π e and π b are the dual weight assigned to productions and the budget constraint. w(v) can be computed recursively in any order implementing the partial order induced by the acyclic productions. As the total column cost according to Koenemann is defined to be the inner product of the column and the dual weights, divided by the objective coefficient, we can get the final tree column cost by adding π dδ to w(t dδ ), and then dividing by c(d δ ). Then among all demands we simply select the one with the smallest tree cost. We traverse the selected tree a second time to obtain the total usages p(e, δ) for each production that the tree uses, which constitute the column coefficients of the tree. We also don t forget the tree s demand production, which has a capacity, and a usage of 1 as column coefficient. The effort to determine the minimum cost trees given dual weights π is O( E ). Young s algorithm [61] finds a feasible point, if it exists, of mixed PLP equations Kx p, Cx c, x 0. (3.10) 1 This does not result in more efficient selection of minimum cost trees, as for different duals traversing the tree for each demand will cover all productions, otherwise the production would not be included. In the time-phased case, it does not apply at all as we always have the sourcing option of taking from inventory vs. from the item production. 2 Problem (3.7) is solved for different values of β a logarithmic number of times

34 22 Chapter 3 Leontief Systems and Extensions for Capacities and Profits in the Deterministic Case To bring (3.3) into conformity here, we move the profit objective into the constraints, requiring a minimum profit of p: (c(d δ ) c(δ)) f(δ) p. (3.11) δ Note that (3.11) has all positive coefficients. This is because if a tree has negative profit (costs exceeding revenue), we do not need to include it at all, as it will not take part in an optimal solution. The reason we can include profit directly here (in contrast to Koenemann s algorithm) is that there is no objective divisor, and the cost of a column j is defined as c(j) := π KK.j π CC.j. (3.12) Of course we still have to do bisection to find the maximum profit p. For given p, the problem then is finding a feasible flow f to (c(d δ ) c(δ)) f(δ) p, δ p(e, δ)f(δ) ū(e), e E, (3.13) e δ f 0. The minimum cost trees can be found by using the same recursion function as before (3.9). The only difference is that instead of dividing by revenue, we subtract π p c(d δ ) (3.14) from the tree cost, where π p is the dual weight for the profit constraint (3.11). The mixed PLP algorithm [61] proceeds in phases during which flow increments are added to all trees with negative cost. As before, to find these trees, essentially a pass through the production matrix is sufficient, requiring O( E ) operations Running Times First we state a result about the mixed PLP feasible point problem. Corollary 3.2. The time to obtain an approximately feasible solution to (3.13) is O(ε 2 F E log E ), where F designates the number of potential bottleneck capacities (in a time-phased formulation, both F and E grow with the number of periods).